Strength functions for collective excitations in deformed nuclei J. Kvasil, 1 N. Lo Iudice, 2 V. O. Nesterenko, 3 and M. Kopa ´ l 1 1 MFF UK, Ke Karlovu 3, Prague, Czech Republic 2 Dipartimento di Scienze Fisiche, Universita ` di Napoli ‘‘Federico II’’ and Istituto Nazionale di Fisica Nucleare, Monte S. Angelo, Via Cinzia I-80126 Napoli, Italy 3 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Received 20 February 1998 A strength function technique valid for a separable Hamiltonian which allows for the coupling between electric and magnetic channels is developed in the framework of proton-neutron random-phase approximation and applied to deformed nuclei. The signature formalism is adopted in view of future applications of the method to fast rotating nuclei. For illustrative purposes the technique is exploited for the computation of the distribution of the magnetic quadrupole strength in axially deformed nuclei, where the coupling with the electric dipole giant resonance is induced by the deformation. S0556-28139801607-0 PACS numbers: 21.10.Re, 21.60.Ev, 21.60.Jz I. INTRODUCTION Strength function techniques are especially valuable for computing the electromagnetic strength distribution in nuclei over an energy range with high level density. By resorting to them, in fact, one can avoid lengthy diagonalization proce- dures and unnecessary detailed calculations for each single state. In spite of these great simplifications, one can reach the same accuracy as one can get in detailed calculations. Dif- ferent strength function techniques have been developed and adopted for different purposes see Refs. 1–4. This paper deals with a technique developed within the random-phase approximation RPA, which adopts a Lorent- zian averaging weight so as to allow for the use of the Cauchy theorem. This technique, which is briefly outlined in Ref. 1, was systematically developed in Refs. 5–8 for computing the electromagnetic strength distribution in heavy nuclei. The procedure was framed in the RPA or in the quasiparticle-phonon nuclear model QPNM9, which ac- counts for the coupling between one and two RPA phonons. A two-body potential of separable form was adopted. The technique was used to compute the strength fragmentation in odd-mass 5,8 as well as in deformed 10,11 and spherical 12 even-mass nuclei. In its original formulation, the method applies only to symmetric secular eigenvalue matrices. On the other hand, in heavy axially deformed even mass nuclei, for a given K quantum number, several multipole fields of both electric and magnetic nature may come into play see, for instance, Ref. 13. When the two-body potential contains more than one multipole field and protons and neutrons are considered as distinct particles, the RPA secular matrix becomes non- symmetric and the strength function method in its original formulation is not applicable. A main purpose of this paper is to develop a symmetriza- tion procedure for the RPA matrices obtained when the separable Hamiltonian includes multipole as well as spin- multipole fields and to show how, once this is done, the strength function can be derived and computed. The whole procedure is developed in the signature formalism which ex- ploits the so-called Goodman basis 14. By this choice, the RPA equations and the strength function method not only become simpler and more transparent, but can be extended, with few modifications, to fast rotating nuclei. The signature formalism is outlined in detail. To our knowledge, in fact, this is the first paper where such a formalism is systemati- cally developed to treat a Hamiltonian of general separable form in RPA and then exploited to compute the strength function. Previous works see, for instance, Ref. 13 used a Hamiltonian which includes multipole and spin-multipole fields but this Hamiltonian was treated in a more conven- tional scheme. Others see, for example, Ref. 15 adopted the signature formalism, but considered a very schematic separable Hamiltonian. For illustrative purposes the method derived here is used to compute the magnetic quadrupole strength function in heavy deformed nuclei. In this case the separable Hamil- tonian contains dipole fields in addition to the dominant spin- dipole terms which enables us to study the influence of the E 1 channel over the M 2 transitions. Some features of the M 2 transitions in deformed nuclei have already been inves- tigated 16,17. In the first calculation 16 rough predictions for the M 2 giant resonance were made. In the other 17, carried out using dipole and spin-dipole fields in the QPNM, only the low-lying spectrum was explored. Unlike the QPNM, we ignore, for the sake of simplicity, the coupling with the two phonons. On the other hand, we adopt the sig- nature formalism. Moreover, the strength function technique enables us to consider not only the low-lying but the full M 2 spectrum, including the region of the M 2 giant resonance. In this respect the two papers complement each other. The paper is organized in the following way. In Sec. II we give the Hamiltonian as well as the single-particle and qua- siparticle basis in the signature formalism. In Sec. III we derive the RPA eigenvalue equations in the same formalism and show how the corresponding matrix can be symmetrized. In Sec. IV we outline the procedure which yields the corre- sponding strength function. Section V deals with the numeri- cal computation of the M 2 strength in the well deformed 154 Sm. Concluding remarks are drawn in Sec. VI. PHYSICAL REVIEW C JULY 1998 VOLUME 58, NUMBER 1 PRC 58 0556-2813/98/581/20911/$15.00 209 © 1998 The American Physical Society